A Fictitious Time Integration Method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization

CMES - Computer Modeling in Engineering and Sciences

Volumn 34, Issue 2, 2008, Pages 155-178

  Liu, Chein Shan ^a   Atluri, Satya N ^b  

^a NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Fictitious time integration method (FTIM); Mixed complementarity problem; NCP functions; Nonlinear optimization problem

Indexed keywords

CAUCHY PROBLEMS; FICTITIOUS TIME INTEGRATION METHOD (FTIM); HIGHLY EFFICIENT; INEQUALITY CONSTRAINTS; LINEAR OPTIMIZATIONS; MIXED COMPLEMENTARITY PROBLEM; NCP-FUNCTIONS; NONLINEAR ALGEBRAIC EQUATIONS; NONLINEAR COMPLEMENTARITY PROBLEMS; NONLINEAR OPTIMIZATION PROBLEM; NOVEL METHODS; NUMERICAL EXAMPLES; NUMERICAL IMPLEMENTATIONS; OPTIMALITY CONDITIONS; OPTIMIZATION PROBLEMS; PLASTICITY EQUATIONS; PRESENT METHODS;

EXPOSURE CONTROLS; FUNCTIONS; INTEGRATION; INVERSE PROBLEMS; OPTIMIZATION; PROBABILITY DENSITY FUNCTION;

NONLINEAR EQUATIONS;

EID: 57649213675     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (49)

1. Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M. (1993): Nonlinear Programming: Theory and Algorithms. Second Edition, John Wiley & Sons, New York.
2. Benson, H. Y.; Shanno, D. F.; Vanderbei, R. J. (2004): Interior-point methods for nonconvex nonlinear programming: jamming and comparative numerical testing. Math. Program., vol. 99, pp. 35-48.
3. Chen, B.; Chen, X.; Kanzow, C. (2000): A penalized Fischer-Burmeister NCP-function. Math. Program., vol. 88, pp. 211-216.
4. Chen, J. S. (2006): The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem. J. Global Optim., vol. 36, pp. 565-580.
5. Chen, J. S. (2007): On some NCP-functions based on the generalized Fischer-Burmeister function. Asia-Pacific J. Oper. Res., vol. 24, pp. 401-420.
6. Chen, J. S.; Gao, H. T.; Pan, S. (2008): A R-linearly convergent derivative-free algorithm for the NCPs based on the generalized Fischer-Burmister merit function, to appear.
7. Chen, J. S.; Pan, S. (2008a): A regularization semismooth Newton method based on the generalized Fischer-Burmeister function for Po-NCPs. J. Comp. Appl. Math., vol. 220, pp. 464-479.
8. Chen, J. S.; Pan, S. (2008b): A family of NCP functions and a descent method for the nonlinear complementarity problem. Comp. Optim. Appl., vol. 40, pp. 389-404.
9. De Luca, T.; Facchinei, F.; Kanzow, C. (1996): A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program., vol. 75, pp. 407-439.
10. Dirkse, S. P.; Ferris, M. C. (1995a): The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optim. Meth. Software, vol. 5, pp. 123-156.
11. Dirkse, S. P.; Ferris, M. C. (1995b): MCPLIB: A collection of nonlinear mixed complementarity problems. Optim. Meth. Software, vol. 5, pp. 319-345.
12. Facchinei, F.; Kanzow, C. (1997): A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Math. Program., vol. 76, pp. 493-512.
13. Facchinei, F.; Soares, J. (1997): A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim., vol. 7, pp. 225-247.
14. Ferris, M. C.; Pang, J.-S. (1997): Engineering and economic applications of complementarity problems. SIAM Review, vol. 39, pp. 669-713.
15. Graham, R. L. (1975): The largest small hexagon. J. Combin. Theory, vol. 18, pp. 165-170.
16. Harker, P. T.; Pang, J.-S. (1990): Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program., vol. 48, pp. 161-220.
17. Ito, K.; Kunisch, K. (2007): On a semi-smooth Newton method and its globalization. Math. Program., Ser. A., DOI 10.1007/s10107-007-0196- 3.
18. Kanzow, C. (2001): Strictly feasible equation-based methods for mixed complementarity problems. Numer. Math., vol. 89, pp. 135-160.
19. Kanzow, C. (2004): Inexact semismooth Newton methods for large-scale complementarity problems. Optim. Meth. Software, vol. 19, pp. 309-325.
20. Jiang, H.; Fukushima, M.; Qi, L.; Sun, D. (1998): A trust region method for solving generalized complementarity problems. SIAM J. Optimization, vol. 8, pp. 140-157.
21. Jiang, H.; Qi, L. (1997): A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim., vol. 35, pp. 178-193.
22. Kuhn, H. W.; Tucker, A. W. (1951): Nonlinear programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Neyman, J. ed., University of California Press, Berkeley, pp. 481-492.
23. Kuo, H. C.; Chang, J. R.; Liu, C. H. (2006): Particle swarm optimization for global optimization problems. J. Marine Sci. Tech., vol. 14, pp. 170-181.
24. Li, S.; Atluri, S. N. (2008a): Topology-optimization of structures based on the MLPG mixed collocation method. CMES: Computer Modeling in Engineering & Sciences, vol. 26, pp. 61-74.
25. Li, S.; Atluri, S. N. (2008b): The MLPG mixed collocation method for material orientation and topology optimization of anisotropic solids and structures. CMES: Computer Modeling in Engineering & Sciences, vol. 30, pp. 37-56.
26. Liu, C.-S. (2001): Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech., vol. 36, pp. 1047-1068.
27. Liu, C.-S. (2008a): Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Problems, vol. 2008, Article ID 749865.
28. Liu, C.-S. (2008b): Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. J. Sound Vib., vol. 318, pp. 148-165.
29. Liu, C.-S. (2008c): A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. CMES: Computer Modeling in Engineering & Sciences, vol. 27, pp. 137-149.
30. Liu, C.-S. (2008d): A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function. CMC: Computers, Materials & Continua, in press.
31. Liu, C.-S. (2008e): A fictitious time integration method for two-dimensional quasi-linear elliptic boundary value problems. CMES: Computer Modeling in Engineering & Sciences, in press.
32. Liu, C.-S. (2008f): A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Eng. Anal. Bound. Elem., vol. 32, pp. 778-785.
33. Liu, C.-S. (2008g): A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains. CMES: Computer Modeling in Engineering & Sciences, vol. 32, pp. 65-75.
34. Liu, C.-S.; Atluri, S. N. (2008): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 31, pp. 71-83.
35. Liu, C.-S.; Chang, J. R.; Chang, K. H.; Chen, Y. W. (2008): Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data. CMC: Computers, Materials & Continua, vol. 7, pp. 97-107.
36. Mangasarian, O. L. (1976): Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math., vol. 31, pp. 89-92.
37. Mangasarian, O. L.; Solodov, M. V. (1993): Nonlinear complementarity as unconstrained and constrained minimization. Math. Program., vol. 62, pp. 277-297
38. Qi, L.; Sun, D. (1993): A nonsmooth version of Newton's method. Math. Program., vol. 58, pp. 353-368.
39. Taji, K.; Miyamoto, M. (2002): A globally convergent smoothing Newton method for nonsmooth equations and its application to complementarity problems. Comp. Optim. Appl., vol. 22, pp. 81-101.
40. Ulbrich, M. (2001): Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optimization, vol. 11, pp. 889-917.
41. Wächter, A.; Biegler, L. (1999): Failure of global converegence for a class of interior point methods for nonlinear programming. Technical Report CAPD Tech Rep B-99-07, Carnegie Mellon University
42. Wang, M. Y.; Wang, X. (2004): PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 6, pp. 373-395.
43. Wang, S. Y.; Wang, M. Y. (2006): Structural shape and topology optimization using an implicit free boundary parametrization method. CMES: Computer Modeling in Engineering & Sciences, vol. 13, pp. 119-147.
44. Wang, S. Y.; Lim, K. M.; Khoo, B. C.; Wang, M. Y. (2007a): A geometric deformation constrained level set method for structural shape and topology optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 18, pp. 155-181.
45. Wang, S. Y.; Lim, K. M.; Khoo, B. C.; Wang, M. Y. (2007b): An unconditionally time-stable level set method and its application to shape and topology optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 21, pp. 1-40.
46. Wang, S. Y.; Lim, K. M.; Khoo, B. C.; Wang, M. Y. (2007c): On hole nucleation in topology optimization using the level set methods. CMES: Computer Modeling in Engineering & Sciences, vol. 21, pp. 219-237.
47. Wang, S. Y.; Lim, K. M.; Khoo, B. C.; Wang, M. Y. (2008): A hybrid sensitivity filtering method for topology optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 24, pp. 21-50.
48. Watson, L. T. (1979): Solving the nonlinear complementarity problem by a homotopy method. SIAM J. Control Optim., vol. 17, pp. 36-46.
49. Yamashita, N.; Fukushima, M. (1995): On stationary points of the implicit Lagrangian for the nonlinear complementarity problems. J. Optim. Theory Appl., vol. 84, pp. 653-663.